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A seismic profiling experiment is performed where the first arrival times of seismic energy from a mid-crustal refractor are observed at distances (in kilometers) ofx=⎣⎡​6.000010.133314.266718.400022.533326.6667​⎦⎤​from the source, and are found to be (in seconds after the source origin time)t=⎣⎡​3.49354.28535.13745.81816.86328.1841​⎦⎤​These vectors can also be found in the MATLAB data file profile.mat. A two-layer, flat Earth structure gives the mathematical modelti​=t0​+s2​xi​,where the intercept time, t0​, depends on the thickness and slowness of the upper layer, and s2​ is the slowness of the lower layer. The estimated noise in the first arrival time measurements is believed to be independent and normally distributed with expected value 0 and standard deviation σ=0.1s.a. Find the least squares solution for the model parameters t0​ and s2​. Plot the data, the fitted model, and the residuals.b. Calculate and comment on the model parameter correlation matrix (e.g., 2.43). How are the correlations manifested in the general appearance of the error ellipsoid in (t0​,s2​) space?c. Plot the error ellipsoid in the (t0​,s2​) plane and calculate conservative 95% confidence intervals for t0​ and s2​ for the appropriate value of Δ2. Hint: The following MATLAB function will plot a two-dimensional covariance ellipse about the model parameters, where C is the covariance matrix, DELTA2 is Δ2, and m is the 2-vector of model parameters.```%set the number of points on the ellipse to generate and plotfunction plot_el1ipse(DELTA2,C,m)n=100;%construct a vector of n equally-spaced angles from ( 0,2*pi)``````theta=1inspace(0,2*pi,n)';%corresponding unit vectorxhat = [cos(theta), sin(theta)];Cinv=inv(C) ;%preallocate output arrayr=zeros(n,2);for i=1:n%store each (x,y) pair on the confidence ellipse%in the corresponding row of }r(i,:)=sqrt(DELTA2/(xhat(i,:)*Cinv*xhat(i,:)'))*xhat(i,:);endplot(m(1) +r(:,1), m(2) +r(:,2));axis equal```d. Evaluate the p-value for this model. You may find the library function chi2cdf to be useful here.e. Evaluate the value of χ2 for 1000 Monte Carlo simulations using the data prediction from your model perturbed by noise that is consistent with the data assumptions. Compare a histogram of these χ2 values with the theoretical χ2 distribution for the correct number of degrees of freedom. You may find the library function chi2pdf to be useful here.f. Are your p-value and Monte Carlo χ2 distribution consistent with the theoretical modeling and the data set? If not, explain what is wrong.g. Use IRLS to find 1-norm estimates for t0​ and s2​. Plot the data predictions from your model relative to the true data and compare with (a) .h. Use Monte Carlo error propagation and IRLS to estimate symmetric 95% confidence intervals on the 1-norm solution for t0​ and s2​.i. Examining the contributions from each of the data points to the 1-norm misfit measure, can you make a case that any of the data points are statistical outliers?

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